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https://github.com/krahets/hello-algo.git
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e720aa2d24
* Sync recent changes to the revised Word. * Revised the preface chapter * Revised the introduction chapter * Revised the computation complexity chapter * Revised the chapter data structure * Revised the chapter array and linked list * Revised the chapter stack and queue * Revised the chapter hashing * Revised the chapter tree * Revised the chapter heap * Revised the chapter graph * Revised the chapter searching * Reivised the sorting chapter * Revised the divide and conquer chapter * Revised the chapter backtacking * Revised the DP chapter * Revised the greedy chapter * Revised the appendix chapter * Revised the preface chapter doubly * Revised the figures
116 lines
3.6 KiB
C++
116 lines
3.6 KiB
C++
/**
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* File: min_path_sum.cpp
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* Created Time: 2023-07-10
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* Author: Krahets (krahets@163.com)
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*/
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#include "../utils/common.hpp"
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/* 最小路径和：暴力搜索 */
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int minPathSumDFS(vector<vector<int>> &grid, int i, int j) {
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// 若为左上角单元格，则终止搜索
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if (i == 0 && j == 0) {
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return grid[0][0];
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}
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// 若行列索引越界，则返回 +∞ 代价
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if (i < 0 || j < 0) {
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return INT_MAX;
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}
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// 计算从左上角到 (i-1, j) 和 (i, j-1) 的最小路径代价
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int up = minPathSumDFS(grid, i - 1, j);
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int left = minPathSumDFS(grid, i, j - 1);
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// 返回从左上角到 (i, j) 的最小路径代价
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return min(left, up) != INT_MAX ? min(left, up) + grid[i][j] : INT_MAX;
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}
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/* 最小路径和：记忆化搜索 */
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int minPathSumDFSMem(vector<vector<int>> &grid, vector<vector<int>> &mem, int i, int j) {
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// 若为左上角单元格，则终止搜索
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if (i == 0 && j == 0) {
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return grid[0][0];
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}
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// 若行列索引越界，则返回 +∞ 代价
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if (i < 0 || j < 0) {
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return INT_MAX;
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}
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// 若已有记录，则直接返回
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if (mem[i][j] != -1) {
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return mem[i][j];
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}
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// 左边和上边单元格的最小路径代价
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int up = minPathSumDFSMem(grid, mem, i - 1, j);
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int left = minPathSumDFSMem(grid, mem, i, j - 1);
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// 记录并返回左上角到 (i, j) 的最小路径代价
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mem[i][j] = min(left, up) != INT_MAX ? min(left, up) + grid[i][j] : INT_MAX;
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return mem[i][j];
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}
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/* 最小路径和：动态规划 */
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int minPathSumDP(vector<vector<int>> &grid) {
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int n = grid.size(), m = grid[0].size();
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// 初始化 dp 表
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vector<vector<int>> dp(n, vector<int>(m));
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dp[0][0] = grid[0][0];
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// 状态转移：首行
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for (int j = 1; j < m; j++) {
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dp[0][j] = dp[0][j - 1] + grid[0][j];
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}
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// 状态转移：首列
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for (int i = 1; i < n; i++) {
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dp[i][0] = dp[i - 1][0] + grid[i][0];
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}
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// 状态转移：其余行和列
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for (int i = 1; i < n; i++) {
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for (int j = 1; j < m; j++) {
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dp[i][j] = min(dp[i][j - 1], dp[i - 1][j]) + grid[i][j];
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}
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}
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return dp[n - 1][m - 1];
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}
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/* 最小路径和：空间优化后的动态规划 */
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int minPathSumDPComp(vector<vector<int>> &grid) {
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int n = grid.size(), m = grid[0].size();
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// 初始化 dp 表
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vector<int> dp(m);
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// 状态转移：首行
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dp[0] = grid[0][0];
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for (int j = 1; j < m; j++) {
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dp[j] = dp[j - 1] + grid[0][j];
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}
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// 状态转移：其余行
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for (int i = 1; i < n; i++) {
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// 状态转移：首列
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dp[0] = dp[0] + grid[i][0];
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// 状态转移：其余列
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for (int j = 1; j < m; j++) {
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dp[j] = min(dp[j - 1], dp[j]) + grid[i][j];
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}
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}
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return dp[m - 1];
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}
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/* Driver Code */
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int main() {
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vector<vector<int>> grid = {{1, 3, 1, 5}, {2, 2, 4, 2}, {5, 3, 2, 1}, {4, 3, 5, 2}};
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int n = grid.size(), m = grid[0].size();
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// 暴力搜索
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int res = minPathSumDFS(grid, n - 1, m - 1);
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cout << "从左上角到右下角的最小路径和为 " << res << endl;
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// 记忆化搜索
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vector<vector<int>> mem(n, vector<int>(m, -1));
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res = minPathSumDFSMem(grid, mem, n - 1, m - 1);
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cout << "从左上角到右下角的最小路径和为 " << res << endl;
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// 动态规划
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res = minPathSumDP(grid);
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cout << "从左上角到右下角的最小路径和为 " << res << endl;
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// 空间优化后的动态规划
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res = minPathSumDPComp(grid);
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cout << "从左上角到右下角的最小路径和为 " << res << endl;
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return 0;
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}
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